On relations between topological and algebraic structures of quasigroups

In this paper we study specific features of the relations between topological and algebraic structures of quasigroups and loops. We study the measurability of subsets of topological quasigroups and loops with respect to invariant measures. We study the family of non-measurable subsets in locally compact non-discrete loops. We find out the existence of locally $\mu $-zero subsets that are not $\mu $-zero in a locally compact left quasigroup that is not $\sigma $-compact. We study quotient spaces of measurable spaces on quasigroups. Moreover, we study homogeneous spaces of quasigroups and countable separability of subsets in them.